Comparison of bipolar vs. tripolar concentric ring electrode Laplacian estimates

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Comparison of bipolar vs. tripolar concentric ring electrode Laplacian estimates Article in Conference proceedings: ... Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Conference · February 2004 DOI: 10.1109/IEMBS.2004.1403656 · Source: PubMed

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Proceedings of the 26th Annual International Conference of the IEEE EMBS San Francisco, CA, USA • September 1-5,2004.

Comparison of Bipolar vs. Tripolar Concentric Ring Electrode Laplacian Estimates W. G. Besio1, R. Aakula2, W. Dai3

1

Department of Biomedical Engineering, Louisiana Tech University-Ruston, LA, USA 2 Department of Electrical Engineering, Louisiana Tech University-Ruston, LA, USA 3 Department of Mathematics & Statistics, Louisiana Tech University-Ruston, LA, USA Abstract— Potentials on the body surface from the heart are of a spatial and temporal function. The 12-lead electrocardiogram (ECG) provides useful global temporal assessment, but it yields limited spatial information due to the smoothing effect caused by the volume conductor. The smoothing complicates identification of multiple simultaneous bioelectrical events. In an attempt to circumvent the smoothing problem, some researchers used a five-point method (FPM) to numerically estimate the analytical solution of the Laplacian with an array of monopolar electrodes. The FPM is generalized to develop a Bi-polar concentric ring electrode system. We have developed a new Laplacian ECG sensor, a Trielectrode sensor, based on a nine-point method (NPM) numerical approximation of the analytical Laplacian. For a comparison, the NPM, FPM and compact NPM were calculated over a 400 x 400 mesh with 1/400 spacing. Tri and Bi-electrode sensors were also simulated and their Laplacian estimates were compared against the analytical Laplacian. We found that Tri-electrode sensors have a muchimproved accuracy with significantly less Relative and Maximum errors in estimating the Laplacian operator. Apart from the higher accuracy, our new electrode configuration will allow better localization of the electrical activity of the heart than Bi-electrode configurations. Keywords—Laplacian, nine point method, Relative error, Maximum error, ECG, EEG, electroencephalography

reported that this special Bi-polar surface Laplacian sensor produced higher resolution maps than BSPMs. This paper will discuss a new Tri-electrode method for measuring surface Laplacian ECG (LECG) that achieves much higher accuracy in duplicating the analytical Laplacian, improved spatial resolution and localization over Bi-polar disc and ring electrode systems. This sensor is based on a numerical approximation technique, the “Nine Point Method” (NPM) which is commonly used in image processing for edge detection. II. BACKGROUND THEORY 1) Five Point Method(FPM): In the Fig. 1 the Laplacian ∆ at point p0 due to the potentials v5,v6,v7,v8 and v0with spacing of 2r [5] is given by

1  4 ∂ 2v ∂ 2v  = ∆ = v +  v − 4v 0  + O(2r ) 2 (1) 0 2 2 2 ∑ i (2r )  i =1 ∂y ∂x  Where 2 4 4 6   4   6 O(r2) = (2r )  ∂ v + ∂ v  + (2r )  ∂ v + ∂ v  + L is 4 4  6 6    4!  ∂x 4!  ∂x ∂y  ∂y  truncation error. This is neglected in [4]. The approximation to the Laplacian at p0 is then

I. INTRODUCTION Body surface potential mapping (BSPM) is a method for improving the spatial resolution of electroencephalography. They can be obtained by recording from a large number of electrodes on the body surface unlike the 12-lead electrode systems. Since BSPMs utilize potentials they also are prone to limited spatial resolution due to the smoothing effect of the volume conductor. The Laplacian of the surface potentials, the second spatial derivative of the surface potentials, may help to sharpen the potentials that were smoothed by the volume conductor effect. In recent studies it was found that body surface Laplacian mapping (BSLM) achieved better spatial resolution in localizing and resolving multiple simultaneously active regional cardiac electrical activity than potential mapping [1]. Fattorusso and Tilmant [2] first reported using the concentric ring electrodes in cardiology. Recently He and Cohen [3] developed a concentric ring Bi-electrode sensor, based on a Five-Point numerical approximation to the analytical solution of Laplacian operator. He and Cohen [4]

∆v 0 ≅ __

where v =

(

4 v − v0 (2r ) 2

)

(2)

1 8 ∑ vi is the average of the potentials on four 4 i =5

points. The above equation can be applied to a concentric disc and ring Bi-electrode sensor by taking the integral along the circle of radius r around the point p0 and defining x= r cos(θ) y= r sin(θ) [5]. Then (2) becomes 2π

4 1 (v(r , θ ) − v 0 )dθ (2r ) 2 2π ∫0 4 ∆v 0 ≅ v − v0 ( 2r ) 2

(3)

∆v 0 ≅

(

__

where

v=

1 2π

)

(4)

2π

∫ v ( r , θ ) dθ 0

potential on the outer ring.

which

is

the

average

2) Compact Nine Point Method(CNPM): The Nine-Point arrangement as shown in Fig. 1 can be interpreted as two FPMs. Points v1,v2,v3,v4 and v0 forming one FPM with a spacing of r and the diagonal points v9,v10,v11,v12 and v0 with a spacing of r forming the second FPM. The Laplacian of the potential at p0 [6] due to these potentials is given by 8  1  4 ∆v0 = 2 4∑ vi + ∑ v j − 20v0  + O(r 2 ) . 6r  i =1 j =5 

∂ v ∂ v 1    2 + 2  = ∆v0 = 2 16∑vi − 60v0 − ∑vi  + O(r 4 ) (6) 12r  i =1 i =5   ∂x ∂y  2

4

8

4 6 6 where O(r )= r  ∂ v + ∂ v  + L is the truncation error. 270  ∂x 6 ∂y 6 

4

Comparing (1) and (5), it can be observed that the NPM truncation error does not have the 4th order derivative term. Therefore the NPM is more accurate than the FPM and also the CNPM.

Y v6

v10

v9

v2 v7

v3

v1

v0 r

v5 r

X

v4 v11

v8

2  2π   (v ( r ,θ )d θ − v ) d θ  = r 2π∆ v 0 0   4 0 

∫

r4 24

+

(5)

3) NPM: The Nine-Point arrangement as shown in Fig. 1 can be seen as two FPMs. Points v1,v2,v3,v4 and v0 forming one FPM with a spacing of r and points v5,v6,v7,v8 and v0 forming a second FPM with spacing of 2r. The Laplacian of the potential at point p0 [6] due to these potentials is 2

electrode sensor, we take the integral along a circle of radius r around point p0 and defining X= rsin(θ), Y= rcos(θ)[5] we get

2π 4

∫ ∑ (sinθ ) 0 j =0

Fig. 1: Arrangement of FPM, CNPM and NPM on a regular plane square grid of size N x N and spacing h =1/N. Inter-point distance r = nh where n=1,2,3,… v0 to v12 are the potentials at points p0 to p12 respectively. v5,v6,v7,v8 and v0 form the FPM , v1,v2,v3,v4, v5,v6,v7,v8 and v0 forming NPM and v1,v2,v3,v4, v9,v10,v11,v12 and v0 forming the CNPM.

III. METHODOLOGY 1) Applying NPM to the New Tri-electrode sensor: As explained before this NPM can be treated as two FPMs. Then by applying a similar procedure as used for the Bi-

 ∂ 4v (cosθ ) j  4− j j  ∂x ∂y

  + L (7) 

Similarly taking the integral along a circle of radius 2r around p0 results in 2π

∫ (v(2r,θ ) − v )dθ = r 0

2

2π∆v0

0

2r 4 + 3 +

( 2r ) 6!

2π 4

 ∂ 4v (sin θ ) 4 − j (cos θ ) j  4− j j  ∂x ∂y  j =0

∫∑ 0

6 2π 6

∫ ∑ (sinθ )

6− j

0 j =0

   

 ∂ 6v  (cosθ ) j  6− j j  + L  ∂x ∂y   

(8)

Combining equations (7) & (8) as {16*(7)-(8)} so as to cancel out the fourth order term, then the Laplacian approximation becomes

  1 2π   v ( r,θ )dθ − v 0   16 ∫ 1   2π 0   ∆v 0 ≅ 2   3r   1 2π    −  2π ∫ v ( 2 r,θ )dθ − v 0   0    where 1 2π

2π

∫ v(r ,θ )dθ

(9)

represents the average potential on

0

the middle ring and v12

4− j

1 2π

2π

∫ v(2r ,θ )dθ

represents the

0

average potential on the outer ring. 2) Comparing Laplacian approximations of NPM, FPM & CNPM: A mesh of 400 x 400 is constructed with a spacing of 1/400. On each and every point of this mesh, both the FPM and NPMs are applied to approximate the Laplacian. This process is repeated for different r values as explained in Fig.1. These estimates are then compared with the calculated analytical Laplacian for each point of the mesh. To calculate the analytical Laplacian, first the electric potentials generated by a dipole in a homogeneous medium of conductivity σ [1] were calculated using the formula for the electrical potential in (10).

φ=

1 4πσ

⋅

(r

p

)

−r ⋅P

(10)

3

rp − r

using (11). The errors are plotted for different inter point distances (r) as shown in Fig. 4. B. Error calculation for Bi & Tri electrode sensors

Where r =(x, y, z) and P =(px, py, pz) represent the location and moment of the dipole, and rp =(xp, yp, zp) represents the observation point. Then the analytical Laplacian was found by taking the second derivative of the potential as below 2 2 L= ∆φ = ∂ φ2 + ∂ φ2

∂x

∂y

 L= 3 5 z − z p 4πσ   

(

) (r − r )⋅ P − (r − r )⋅ P + 2(z 2

p

p

rp − r

7

rp − r

5

p

 − z pz  .   

)

(11)

Relative error and Maximum errors of Bi-electrode and Tri-electrode sensors are tabulated in Table I. Two sample t-tests which assumed unequal variance were used to test significance. The Relative errors of the NPM were compared both between the FPM and the CNPM. The Maximum errors of the NPM were compared similarly. It was found that the NPM was significantly better in all four cases at the 1% confidence level. The mean percent improvement of error by the NPM compared to the other methods ranged from 99.65% to 99.88%.

The Laplacian approximations of the two finite difference methods are compared with those of the analytical by calculating the Relative Error and Maximum Error [7].  RELERRi =   

∑ (∆v − ∆ v) ∑ (∆v) i

2

2

n

n

1

2   

i

MAXERR = max ∆v − ∆i v

(12) Fig. 2: Tri-electrode sensor, n is inter-electrode distance

(13) Tri-electrode sensor

Z

where i represents the method used to find the Laplacian and ∆v represents the analytical Laplacian of the potential. 3) Comparing Laplacian approximation of Bielectrode and Tri-electrode sensor: The Tri-electrode sensor as shown in Fig. 2 can be treated as either a Bi-electrode or a Tri-electrode sensor by ignoring (utilizing) the dashed ring for Bi-electrode (Tri-electrode) calculations. This Trielectrode sensor is simulated at a particular height above the origin as shown in Fig. 3 by taking 0.01 times 2*pi as the angular displacement between each point on the rings. A dipole is simulated which is oriented to the surface in the positive direction of the Z-axis as shown in Fig. 3. Using this Tri & Bi-electrode configuration, the Laplacian is calculated at the center with varying interelectrode distance (n) and with a fixed dipole at the origin. These Laplacian values are compared with the analytical values by calculating the Relative Error and Maximum Error [7]. IV. RESULTS A. Error calculation for the NPM, FPM & CNPM Relative and Maximum errors of the NPM, FPM and CNPM are calculated using formulae (12) and (13) when compared to the analytical value of Laplacian calculated

n

n

Dipole

X Y Fig. 3: Schematic of simulation used for Bi-electrode and Tri-electrode sensors at a particular height above the origin with fixed dipole at the origin.

Table I Relative and Maximum errors for Bi and Tri-electrode sensors with various radii Radius n - units 0.5 1 1.5 2 2.5 3

Bi-electrode Sensor Rel. error Max. error 1.91813e-2 4.49563e-4 7.28031e-2 1.70632e-3 1.50892e-1 3.53653e-3 2.41444e-1 5.65885e-3 3.3405e-1 7.82936e-3 4.21629e-1 9.88194e-3

Tri-electrode Sensor Rel. error Max. error 8.70885e-5 2.04113e-6 1.30743e-3 3.06431e-5 5.98458e-3 1.40263e-4 1.65892e-2 3.88811e-4 3.47052e-2 8.13403e-4 6.06464e-2 1.42140e-3

VI. CONCLUSIONS The Tri-electrode configuration gives a significantly better approximation to the analytical Laplacian than the other finite difference methods commonly used. This should be helpful in localizing sources, which will be our future work. We will also conduct tank experiments to verify our simulation results. With an array of these Trielectrode sensors, Laplacian surface potential maps should be made more accurately than past mapping efforts. ACKNOWLEDGMENT (a)

We would like to thank the Louisiana Center for Entrepreneurship and Information Technology (CEnIT) and the Louisiana Board of Regents (grant # LEQSF (2003-05)RD-B-05) and all the help from our lab members and friends. REFERENCES [1] [2] [3] [4] [5]

(b) Fig. 4: Relative Error (a) Maximum Error (b) of NPM, FPM and Compact NPM when compared to the analytical value of the Laplacian

V. DISCUSSIONS The results of the simulations for the Maximum and Relative error of the NPM, FPM and CNPM are plotted on a semi-log graph as shown in Fig. 4 and Bi-electrode and Trielectrode data are tabulated in Table 1. The FPM and CNPM both have truncation error on the order of r2 where as the NPM has order of r4. Hence, we expected the NPM to be more accurate than FPM and CNPM, which is proven by our simulations. The two sample t-tests results showed there was a significant difference in error approximating the Laplacian between the NPM and both of the other finite difference methods compared and between Tri and Bielectrode configurations as well. The NPM is generalized to approximate the Laplacian for the Tri-electrode sensor. We expected the Tri-electrode sensor to be more accurate than the Bi-electrode sensor due to the difference in truncation error, and it is proven so by the simulation results.

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B. He and D. Wu, “Laplacian electrocardiography,” Critical Reviews TM in Biomedical Engineering, 27(3-5):285-338 (1990). V Fattorusso and J Tilmant, “Exploration du champ electrique precordial a l’aide de deux electrodes circulaires, concentriques et rapprochees,” Arch. Mal du Coeur. 42, pp. 452-455, 1949. B He, and RJ Cohen, “Body surface Laplacian ECG mapping,” IEEE Trans. on BME. 39(11), pp. 1179-1191, 1992. B He, and RJ Cohen, “Body surface Laplacian mapping in Man,” IEEE EMBS 13(2), pp. 784-786,1991. Leon Lapidus and Geroge F. Pinder, Numerical solution of partial differential equations in science and engineering, New York: John Wiley & Sons, Inc. 1982, pp.371-372. William F. Ames, Numerical methods for partial differential equations, New York: Barnes & Noble, Inc., 1969, pp. 15-19 Geertjan Huiskamp, ”Difference formulas for the surface Laplacian on a triangulated surface,” Journal of Computational Physics, 95(2), pp.477-496, 1991.